On the Optimal Wealth Process in a Log-Normal Market: Applications to Risk Management
34 Pages Posted: 30 Apr 2014 Last revised: 31 Jul 2015
Date Written: June 5, 2014
Abstract
Using a stochastic representation of the optimal wealth process in the classical Merton problem, we calculate its cumulative distribution and density functions and provide bounds and monotonicity results for these quantities under general risk preferences. We also show that the optimal wealth and portfolio processes for different utility functions are related through a deterministic transformation and appropriately modified initial conditions. We analyze the value at risk (VaR) and expected shortfall of the optimal wealth process and show how each can be used to infer a CRRA investor's risk aversion coefficient. Drawing analogies to the option greeks, we study the sensitivities of the optimal wealth process with respect to the cumulative excess stock return, time, and market parameters. We conclude with a study of how sensitivities of the excess return on the optimal wealth process relate to the portfolio's beta.
Keywords: expected utility, Merton problem, value at risk, expected shortfall, portfolio Greeks
JEL Classification: G10, G11
Suggested Citation: Suggested Citation